Phd Students:

About the Department

The full time employees in the past few years have been: Kazimierz Alster,
Józef Krasinkiewicz, Wiesław Olędzki, Stanisław Spież and Henryk Toruńczyk.
Moreover, within the last 3 years the following topologists have been
or are currently employed for a period varying from 6 months to
2 years:
T. Dobrowolski, K. Gęba, G. Graff, M. Izydorek,
T. Januszkiewicz, J. Jezierski, D. Kołodziejczyk,
T. Koźniewski, S. Nowak, R. Mańka, J. Świątkowski, A. Tralle,
P. Traczyk.

Below are the descriptions of the main research results or
research areas of the full time employees and the list of some of the
papers published (either containing the results discussed or
completed within the last 3 years).

Kazimierz Alster

My research area belongs to General Topology. I have been interested
in studying the properties of a space which can be expressed in terms
of its coverings. These include: the Hurewicz property, the Eberlein
compactness, the Fréchet property, paracompactness, and the Lindelöf
property of Cartesian products. The study of these properties
often leads to problems close to infinite combinatorics.

Józef Krasinkiewicz

My main results include:

Showing in [11] that if every map of a compactum X into
R2n can be approximated by an embedding then
dim (X×X) < 2n.
(The reverse implication and other extensions have subsequently been obtained
by many authors, primarily by S. Spież and by H. Toruńczyk.)

Proving, jointly with Z. Karno [9], that the basic property
dim(X×X) < 2 dim(X)
celebrated Boltyanskii-Kodama compacta and of their natural
n-dimensional counterparts can be established using the above result
by studying the set of all mappings of these spaces into R2n.
(This research grew out from a joint paper with K. Lorentz in
Bull. Polish Acad. Sci. 36 (1988), 397-402.)

Proving that mappings of a compact metric space into manifolds can
be approximated by mappings whose fibers are hereditarily indecomposable,
and that in fact the set of such maps is a dense Gδ
(see [12]).

Extending the notion of an essential mapping into spheres
to the case when the target space is a product of manifolds.
This new notion has been applied (see [10])
to construct spaces with peculiar properties (e.g.: there exists
a nondegenerate continuum whose each subset of positive dimension
admits an essential map onto each sphere) and to proving
decomposition theorems
(e.g.: if the Hilbert cube Q is expressed as a countable union of sets,
then one of them contains a connected non-trivial subset, which
strengthens Hurewicz's result on decompositions of Q into
zero-dimensional sets).

Stanisław Spież

My major areas of research are shape theory (and indirectly homotopy
theory), dimension theory, theory of embeddings and game theory.

Several of my papers in shape theory are devoted to studying movable
spaces (their role
is similar to that of CW-complexes in homotopy theory)
and deformation dimension (which corresponds to the homotopical
dimension).
Some of my results in
that area are related to the classical Whitehead
and Hurewicz theorems in homotopy theory.
Also I investigated the possibility of representing the strong
shape category in the homotopy category.

Another area of my research
is related to the following questions:
"When can a pair of mappings of compact metric spaces X
and Y into
Rn be approximated by mappings
with disjoint images, and also when can a map X→Rn be
approximated by embeddings?'' Since the 1930's
the standard answer
to the latter has been "It suffices that 2 dim X < n"
turns out that it is sufficient that dim(X×X) < n.
Some other papers of mine concern the first question (which is more general).

Also I was interested in the questions of embedding polyhedra into
Euclidean spaces, which were related to the van Kampen and Haefliger-Weber
theorems.

Recently I am also involved in research in game theory.
Some results on the existence of equilibria in a class of games
can be proved by using topological tools, such as
coincidence theorems of Borsuk-Ulam type.

Several of the above results were obtained
in collaboration
with the following mathematicians: B. Günther,
S. Nowak, J. Segal, R. Simon, A. Skopenkov and H. Toruńczyk.

Henryk Toruńczyk

Major part of my research concerned topological properties of
infinite-dimensional spaces, such as the Hilbert cube or Banach
spaces. I consider the following my main results:

developing a method of constructing smooth partitions of unity on
Banach spaces in the absence of separability [20];

proving that a product of an absolute retract with an
appropriate normed linear space becomes homeomorphic to that
space [21];

proving, simultaneously with S. Ferry, that the homeomorphism
group of a Hilbert cube manifold is a manifold [22];

characterizing infinite-dimensional manifolds topologically
(see R. D. Edwards' article in SLN 770, 278-302).
As a consequence it turned out that infinite-dimensional Banach
spaces of the same weight are homeomorphic;

examining, jointly with S. Spież, when mappings
X, Y→Rk can be ε-approximated by mappings
with disjoint images [18];

establishing, jointly with R. Simon and S. Spież,
the existence of equilibria in a class of infinitely repeated games.
(The proof in [19] depended on developing an appropriate
topological aparatus.)